Friday, September 30, 2011

"A Phenomenological Perspective,"
Ch. 2 in The Star and the Whole:
Gian-Carlo Rota on Mathematics and Phenomenology ,
by Fabrizio Palombi, A K Peters/CRC Press, 2011—

"Rota is convinced that one of the fundamental tasks of phenomenology is that of highlighting the primordiality of sense. In his words, if 'many disputes among philosophers are disputes about primordiality' then 'phenomenology is yet another dispute about what is most primordial' (Rota, 1991a,* p. 54). In this way he evidently does not intend to deny the existence of matter, of objects, or of that objective dimension proper to science, in favor of a spiritualist option, but rather to posit as primordial another dimension of the world connected with contexts and with roles, which is considered primordial because each one of us is confronted with it primordially."

* The End of Objectivity: The Legacy of Phenomenology ,
Lectures by Rota at MIT 1974-1991, 457 pages,
MIT Mathematics Department, Cambridge, MA

"In a paper in the current issue of Nature , Dr. Lloyd describes the ultimate laptop— a computer as powerful as the laws of physics will allow. So energetic is this imaginary machine that using it would be like harnessing a thermonuclear reaction. In the most extreme version of this computer supreme, so much computational circuitry would be packed into so small a space that the whole thing would collapse and form a tiny black hole, an object so dense that not even light can escape its gravity."

"We wish to see Jesus. For somehow we know, we suspect, we intuit, that if we see Jesus we will see what Meister Eckhart might call “The Divine Kernel of Being”— that Divine Spark of God’s essence, God’s imago Dei, the image in which we are created. We seem to know that in seeing Jesus we just might find something essential about ourselves."

—The Reverend Kirk Alan Kubicek, St. Peter’s at Ellicott Mills, Maryland, weblog post of Saturday, March 28, 2009, on a sermon for Sunday, March 29, 2009

The title describes two philosophical events (one major, one minor) from the same day— Thursday, July 5, 2007. Some background from 2001:

"Are the finite simple groups, like the prime numbers, jewels strung on an as-yet invisible thread? And will this thread lead us out of the current labyrinthine proof to a radically new proof of the Classification Theorem?" (p. 345)

The major event— On July 5, 2007, Cambridge University Press published Robert T. Curtis's Symmetric Generation of Groups.*

Curtis's book does not purport to lead us out of Solomon's labyrinth, but its publication date may furnish a Jungian synchronistic clue to help in exiting another nightmare labyrinth— that of postmodernist nominalism.

Bridget Fonda in "Point of No Return." This is a 1993 remake of 1990's "La Femme Nikita ,"
virtually the same scene-by-scene, but with two nice new touches: the young assassin's code name
is Nina, after Nina Simone, and she happily shops as Simone sings "new day" in the soundtrack.

"… we are saying much more than that G ≅M 24 is generated by
some set of seven involutions, which would be a very weak
requirement. We are asserting that M 24 is generated by a set
of seven involutions which possesses all the symmetries of L3(2)
acting on the points of the 7-point projective plane…."
— Symmetric Generation , p. 41

"It turns out that this approach is particularly revealing and that
many simple groups, both sporadic and classical, have surprisingly
simple definitions of this type."
— Symmetric Generation , p. 42

See also (click to enlarge)—

Cassirer's remarks connect the concept of objectivity with that of object .

The above quotations perhaps indicate how the Mathieu group M 24 may be viewed as an object.

"This is the moment which I call epiphany. First we recognise that the object is one integral thing, then we recognise that it is an organised composite structure, a thing in fact: finally, when the relation of the parts is exquisite, when the parts are adjusted to the special point, we recognise that it is that thing which it is. Its soul, its whatness, leaps to us from the vestment of its appearance. The soul of the commonest object, the structure of which is so adjusted, seems to us radiant. The object achieves its epiphany."

— James Joyce, Stephen Hero

For a simpler object "which possesses all the symmetries of L3(2) acting on the points of the 7-point projective plane…." see The Eightfold Cube.

…. A note of Feb. 20, 1986, supplied an example of such coordinatizations in finite geometry. In that note, the group of mediating transformations acted directly on coordinates within a 4×4 array. When the 4×4 array is embedded in a 4×6 array, a larger and more interesting group, M 24 (containing the original group), acts on the larger array. There is no obvious solution to Weyl’s relativity problem for M 24. That is, there is no obvious way* to apply exactly 24 distinct transformable coordinate-sets (or symbol-strings ) to the 24 array elements in such a way that the natural group of mediating transformations of the 24 symbol-strings is M 24. ….

Footnote of Sept. 20, 2011:

* R.T. Curtis has, it seems, a non-obvious way that involves strings of seven symbols. His abstract for a 1990 paper says that in his construction “The generators of M 24 are defined… as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters….”

"… it seems that the relationship between
BIB [balanced incomplete block ] designs
and tactical configurations, and in particular,
the Steiner system, has been overlooked."
— D. A. Sprott, U. of Toronto, 1955

The figure by Cullinane included above shows a way to visualize Sprott's remarks.

A different illustration of the eightfold cube as the Steiner system S(3, 4, 8)—

Note that only the static structure is described by Felsner, not the
168 group actions discussed (as above) by Cullinane. For remarks on
such group actions in the literature, see "Cube Space, 1984-2003."

"Now suppose that α is an element of order 23 in M 24 ; we number the points of Ω
as the projective line ∞, 0, 1, 2, … , 22 so that α : i→i + 1 (modulo 23) and fixes ∞. In
fact there is a full L 2 (23) acting on this line and preserving the octads…."

Those who prefer, as Weyl put it,
"the hard core of objectivity"
to, as Eddington put it,
"the colorful tale of the subjective storyteller mind"
may consult this journal on the same day… July 18, 2008.

"'Ain’t No Grave' is Johnny’s final studio recording. The album and its title track
deal heavily with themes of mortality, resurrection, and everlasting life.The Johnny Cash Project pays tribute to these themes."

A user wanting to custom-tailor his self-symbol should consider
the following from the identicon service Gravatar—

1. User Submissions. "… you hereby do and shall grant to Automattic a worldwide, perpetual, irrevocable, royalty-free and fully-paid, transferable (including rights to sublicense) right to perform the Services (e.g., to use, modify, reproduce, distribute, prepare derivative works of, display, perform, and otherwise fully exercise and exploit all intellectual property, publicity, and moral rights with respect to any User Submissions, and to allow others to do so)."

The JSTOR journal archive announced today that it is making nearly 500,000 public domain journal articles from more than 220 journals—or about six percent of JSTOR's total content—freely available for use by "anyone, without registration and regardless of institutional affiliation."

The material, entitled Early Journal Content, will be rolled out in batches starting today over the course of one week. It includes content published in the United States before 1923 and international content published before 1870, which ensures that all the content is firmly in the public domain. JSTOR, in an announcement, said that the move was "a first step in a larger effort to provide more access options" to JSTOR content for independent scholars and others unaffiliated with universities.

"This won't be just another upgrade. Windows 8 is nothing less than the linchpin to Microsoft's strategy for keeping Windows relevant— if not resurgent— as the shift to the post-PC computing era unfolds.

'The stakes are huge,' says Charles King, principal analyst at research firm Pund-IT. 'The company must play outside its comfort zone, but if Microsoft succeeds, the potential opportunities could be significant.'"

She drew from her handbag a pale grey gleaming implement
that looked by quick turns to me like a knife, a gun,
a slim sceptre, and a delicate branding iron— especially when
its tip sprouted an eight-limbed star of silver wire.

“The test?” I faltered, staring at the thing.

“Yes, to determine whether you can live in the fourth dimension or only die in it.”

Coxeter later uses the the 3×3 array (with center omitted) again to illustrate the Desargues configuration—

The Desargues configuration is discussed by Gian-Carlo Rota on pp. 145-146 of Indiscrete Thoughts—

"The value of Desargues' theorem and the reason why the statement of this theorem has survived through the centuries, while other equally striking geometrical theorems have been forgotten, is in the realization that Desargues' theorem opened a horizon of possibilities that relate geometry and algebra in unexpected ways."

Comments Off on Starring the Diamond

Wednesday, September 7, 2011

A search for some background on Gian-Carlo Rota's remarks
in Indiscrete Thoughts * on a geometric configuration
leads to the following passages in Hilbert and Cohn-Vossen's
classic Geometry and the Imagination—

These authors describe the Brianchon-Pascal configuration
of 9 points and 9 lines, with 3 points on each line
and 3 lines through each point, as being
"the most important configuration of all geometry."

The Encyclopaedia of Mathematics , ed. by Michiel Hazewinkel,
supplies a summary of the configuration apparently
derived from Hilbert and Cohn-Vossen—

My own annotation at right above shows one way to picture the
Brianchon-Pascal points and lines— regarded as those of a finite,
purely combinatorial , configuration— as subsets of the nine-point
square array discussed in Configurations and Squares. The
rearrangement of points in the square yields lines that are in
accord with those in the usual square picture of the 9-point
affine plane.

A more explicit picture—

The Brianchon-Pascal configuration is better known as Pappus's configuration,
and a search under that name will give an idea of its importance in geometry.

"She has taken on the role of a public face of physics,
and has written a book which is in part a very general defense
of science and the materialist, rationalist world-view
that modern science is based on."

"Dan Brown certainly packed a lot into the 500-plus pages of The Lost Symbol . But perhaps the key element to the story is the search for the ‘Lost Word,’ and— in the final pages— Robert Langdon’s discovery as to what that actually means. In the early chapters, Langdon explains to Sato that the Lost Word was 'one of Freemasonry’s most enduring symbols'…

…a single word, written in an arcane language that man could no longer decipher. The Word, like the Mysteries themselves, promised to unveil its hidden power only to those enlightened enough to decrypt it. “It is said,” Langdon concluded, “that if you can possess and understand the Lost Word . . . then the Ancient Mysteries will become clear to you.”

Saturday, September 3, 2011

A post of September 1, The Galois Tesseract, noted that the interplay
of algebraic and geometric properties within the 4×4 array that forms
two-thirds of the Curtis Miracle Octad Generator (MOG) may first have
been described by Cullinane (AMS abstract 79T-A37, Notices , Feb. 1979).

Here is some supporting material—

The passage from Carmichael above emphasizes the importance of
the 4×4 square within the MOG.

The passage from Conway and Sloane, in a book whose first edition
was published in 1988, makes explicit the structure of the MOG's
4×4 square as the affine 4-space over the 2-element Galois field.

The passage from Curtis (1974, published in 1976) describes 35 sets
of four "special tetrads" within the 4×4 square of the MOG. These
correspond to the 35 sets of four parallel 4-point affine planes within
the square. Curtis, however, in 1976 makes no mention of the affine
structure, characterizing his 140 "special tetrads" rather by the parity
of their intersections with the square's rows and columns.

The affine structure appears in the 1979 abstract mentioned above—

The "35 structures" of the abstract were listed, with an application to
Latin-square orthogonality, in a note from December 1978—

“In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an ‘octad generator’; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code.”

Friday, September 2, 2011

"… If the cliffhanger is done well, it will not
simply introduce a wholly unprepared turn
into the narrative (a random death, a new character,
an entirely unanticipated obstacle) but rather
tighten the configuration of known elements…."

"Today, Mazur says he has woken up to the power of narrative, and in Mykonos gave an example of a 20-year unsolved puzzle in number theory which he described as a cliff-hanger. 'I don’t think I personally understood the problem until I expressed it in narrative terms,' Mazur told the meeting. He argues that similar narrative devices may be especially helpful to young mathematicians…."

… If the cliffhanger is done well, it will not simply introduce a wholly unprepared turn into the narrative (a random death, a new character, an entirely unanticipated obstacle) but rather tighten the configuration of known elements to such a degree that the next step appears both inevitable and impossible. We feel the pressure rising to a breaking point, but we simply cannot foresee where the complex narrative structure will give way. This interplay of necessity and contingency produces our anxious— and highly pleasurable— speculation about the future path of the story. But if we could determine that path even slightly, we would narrow the range of possible outcomes and thus the uncertainty in the play of necessity and contingency. The world of the fiction would feel, not open, but rigged."

* The idea of the thought experiment emerged in a conversation with Barry Mazur.

"But the telltale adjective real suggests two things: that these numbers are somehow real to us and that, in contrast, there are unreal numbers in the offing. These are the imaginary numbers .

The imaginary numbers are well named, for there is some imaginative work to do to make them as much a part of us as the real numbers we use all the time to measure for bookshelves.

This book began as a letter to my friend Michel Chaouli. The two of us had been musing about whether or not one could 'feel' the workings of the imagination in its various labors. Michel had also mentioned that he wanted to 'imagine imaginary numbers.' That very (rainy) evening, I tried to work up an explanation of the idea of these numbers, still in the mood of our conversation."

"By far the most important structure in design theory is the Steiner system S(5, 8, 24)."
— "Block Designs," by Andries E. Brouwer

The name Carmichael is not to be found in Booher's thesis. In a reference he does give for the history of S(5,8,24), Carmichael's construction of this design is dated 1937. It should be dated 1931, as the following quotation shows—

"The linear fractional group modulo 23 of order 24•23•11 is often represented as a doubly transitive group of degree 24 on the symbols ∞, 0, 1, 2,…, 22. This transitive group contains a subgroup of order 8 each element of which transforms into itself the set ∞, 0, 1, 3, 12, 15, 21, 22 of eight elements, while the whole group transforms this set into 3•23•11 sets of eight each. This configuration of octuples has the remarkable property that any given set of five of the 24 symbols occurs in one and just one of these octuples. The largest permutation group Γ on the 24 symbols, each element of which leaves this configuration invariant, is a five-fold transitive group of degree 24 and order 24•23•22•21•20•48. This is the Mathieu group of degree 24."

"While all this [Captain America] is happening an SS officer, Johann Schmidt (Hugo Weaving), has found a religious artefact called the Tesseract which Schmidt describes as 'the jewel of Odin’s treasure room,' linking it in with the Thor storyline."

— That's Entertainment weblog, August 14, 2011

From Wallace Stevens, "An Ordinary Evening in New Haven," Canto III—

The point of vision and desire are the same.
It is to the hero of midnight that we pray
On a hill of stones to make beau mont thereof.